What is Algebraic Geometry?
The University of Utah well known for its specialization in the field of Algebraic Geometry. It has attracted many professors and graduate students to
come research or attend one of the numerous conferences on the subject. For
these reasons it is important for first year graduate students to learn some of
the basic principles of Algebraic Geometry. On January 19, 26 and February 2,
2007 professors Aaron Bertram, Christopher Hacon, and Tommaso de Fernex
presented introductory concepts and general insights into their research specialties for the pilot course of Early Research Directions. Transcribing these
lectures were (in order), Erika Meucci, Peter Marcy, and Liang Zhang.
Aaron Bertram
Algebraic Geometry is the study of the solutions to systems of polynomial
equations.
As for the algebraic aspects, there is a natural question which comes up:
Which fields are to be used for coefficients and for the solutions? For example,
we can consider fields like Q, Fp , R, or Q(t) which are not algebraically closed;
or we could take Q, Fp or C, which are, of course, algebraically closed. We can
choose one field for the coefficients, say Q, and another field for the solutions,
say C. In particular, R and C are called geometric fields because we are able
to visualize them.
Let k be the chosen field. An (k) = k n is called the affine space over k.
Now we consider the polynomials f1 , . . . , fr ∈ k[x1 , . . . , xn ]. We want to study
the set
V (f1 , . . . , fr ) = (f1 = · · · = fr = 0) ⊂ An (k).
Another space that is very important is the projective space. The projective space is a compactification of An (k). We can think, for example, k = R
or C. How does it work? We can think Pn (k) as the set of the lines through
the origin in An+1 (k). For example, P1 (R) is the set of the lines through the
origin in R2 . If we intersect these lines with the line y = 1, we have that the
intersection is a point for each line except for the line y = 0. But if you consider
the intersection between these lines and x = 1, we have that the intersection is
a point for each line except for the line x = 0. Hence, if Uy=1 is the set of the
lines through the origin which intersect y = 1 and Ux=1 is the set of the lines
through the origin which intersect x = 1, then
S 1 = P1 (R) = A1 (R) ∪ {∞} = Ux=1 ∪ Uy=1 .
This works in general, i.e.
P2 (k) = A2 (k) ∪ P1 (k) = Ux=1 ∪ Uy=1 ∪ Uz=1 ,
1
where we think P1 (k) as the set of the points at infinity, that is the lines in the
xy-plane.
The basic fact is that if we have some polynomial equations (f1 = · · · =
fr = 0) ⊂ An (k), then there exist homogenous polynomials F1 , . . . , Fs such that
(F1 = · · · = Fs = 0) ⊂ Pn (k) is equal to the closure of (f1 = · · · = fr = 0). For
example, the homogeneous polynomial associated to y 2 = x+a is y 2 = xz +az 2 .
Since (F1 = · · · = Fs = 0) is a closed subset of the compact set Pn (k),
(F1 = · · · = Fs = 0) is compact, and this is a very important fact in Algebraic
Geometry.
Important concepts come from the conic sections. We consider the cone
F = (x2 + y 2 = z 2 ) and we are interested in the intersection between this cone
and a plane. If we consider the plane Uz=1 , we have
(F = 0) ∩ Uz=1 = (x2 + y 2 = 1)
which is S 1 . If we consider the plane Uy=1 , we have
(F = 0) ∩ Uy=1 = (x2 − z 2 = 1)
which is a hyperbola.
Given f1 , . . . , fr ∈ k[x1 , . . . , xn ], we can form the ideal generated by these
polynomials:
hf1 , . . . , fr i :=
r
nX
o
gi fi | gi ∈ k[x1 , . . . , xn ] .
i=1
The following Theorems are among the most important in Algebraic Geometry.
Theorem 1 (Hilbert Basis Theorem) Any ideal in k[x1 , . . . xn ] is of the
form < f1 , . . . , fr >.
Theorem 2 (Hilbert Nullstellensatz) If k is algebraically
closed, then
p
(f
=
· · · = fr = 0) =
(f
=
·
·
·
=
f
=
0)
=
(h
=
·
·
·
=
h
=
0)
if
and
only
if
1
1
r
1
s
p
(h1 = · · · = hs = 0).
p
In the above theorem (f1 = . . . = fr = 0) is the radical ideal of (f1 = · · · =
fr = 0), and is defined to be {g ∈ k[x1 , . . . , xn ] | ∃N : g N ∈ hf1 , . . . , fr i} .
The building blocks of this theory are the prime ideals. An ideal P ⊂
k[x1 , . . . xn ] is prime if the quotient k[x1 , . . . xn ]/P is a domain. Another important notion is that of dimension. The dimension, denoted by dim(P = 0),
is the transcendence degree of the field of fractions of k[x1 , . . . , xn ]/P . We call
X := (P = 0) and k(x) the field of fractions of k[x1 , . . . , xn ]/P .
2
A very difficult problem in Algebraic Geometry is the following: given P ,
determine if k(x) = k(t1 , . . . , tdim(P =0) ). There are some results about this
problem, but we don’t have a general answer. If there are more than three
equations or more variables than one, the problem is still open.
And now we move on to Geometry. We consider a homogeneous prime ideal Q.
What are the (global) invariants of X = (Q = 0) ⊂ Pn (k)?
What is the meaning of a function being smooth at one point? A function is smooth at one point x0 if it is infinitely many differentiable in x0 . For
example, y 2 = x2 (x + 1) and y 2 = x3 are not smooth at the origin.
We will try to understand what happens in general. We define (P = 0) :=
(Q = 0) ∪ Ux=1 ⊂ An (k) and we think (P = 0) as the same to see (Q = 0)
locally. One of the most powerful Theorems in Algebraic Geometry (and in
many other branches of mathematics) is the Implicit Function Theorem:
Theorem 3 (Implicit Function Theorem)
If k = R or C and there are
∂fi
f1 , . . . , fn−d ∈ P such that the matrix ∂xj has rank n−d at p ∈ X = (P = 0),
then X has analytic local coordinates at p.
For example, x is a nice analytic local coordinate of x2 + y 2 = 1 near (0, 1).
We do not want an ramified map so if we look at a neighborhood, we have
an analytic function, but if we look at this function globally, we don’t have an
analytic function. Hence, we need to develop a new theory, that is Algebraic
Geometry.
I would like to explain how I got into Algebraic Geometry. I wanted to
solve the Fermat equation xn + y n = z n . The statement of Fermat’s Last
Theorem is the following:
Theorem 4 (Fermat’s Last Theorem) There are not solutions in Z for
xn + y n = z n when n > 2, except the obvious solutions.
We can think of (xn + y n − z n = 0) in P2 (Q) or in P2 (C) and we can consider
the graph
(xn + y n = 1) = (xn + y n = z n ) ∩ Uz=1 .
Obviously, if n = 1, we get a line and we have infinitely many solutions in
Q. If n = 2, we have infinitely many solutions and we can find the solutions by
considering the parametrization
2t
t2 − 1 ,
,
t2 + 1 t2 + 1
t ∈ Q.
Moreover, if Cn (k) = (xn +y n −z n = 0) ⊂ P2 (C), we have a one-dimensional
compact complex manifold (called a Riemann Surface). Given Q ∈ Q[x1 , . . . , xn+1 ]
3
homogeneous, prime ideal we call C(Q) = (Q = 0) ⊂ Pn (C) a Riemann Surface
(i.e. a smooth projective algebraic curve).
One-dimensional compact complex manifolds are classified by an invariant
called the genus g (or the numbers of holes), which is equal to
(n − 1)(n − 2)
.
2
For example, if n = 1, then g = 0, so we have a sphere. If n = 2, then g = 0.
If n = 3, then g = 1; that is a torus. If n = 4, then g = 3, and so on. There are
three fundamental Theorems about the genus of Riemann Surfaces:
g=
Theorem 5 If g = 0, then C(Q) is empty or there exists a 1−1 correspondence
between C(Q) and Q ∪ {∞} = P1 (Q).
Theorem 6 (Mordell) If g = 1, then C(C) is an abelian group and C(Q) is
a finitely generated abelian group.
Theorem 7 (Faltings, ∼ 1980’s) If g ≥ 2, then C(Q) is finite.
A question of paramount importance is “How we can generalize these Theorems
to higher dimensions?”
Christopher Hacon
My primary interest is the “Birational Classification of Complex Projective
Algebraic Varieties”. That is, X ⊂ PN
C is defined by homogeneous polynomials
−1
P1 , ..., Pt and PN
is
the
compactification
of CN = CN ∪ PN
. An example of
C
C
1
this compactification is PC = C ∪ {∞}.
Two varieties X1 , X2 are birational if they have isomorphic non-empty open
subsets; in other words a set of measure zero can be thrown away to get an isomorphism. Some examples of smooth 1-dimensional varieties, X depend on the
genus g of a surface. For g = 0, there is only one variety: the Riemann sphere.
For g = 1, there is only one surface topologically, but there is a 1-dimensional
family of varieties to study. Similarly for g ≥ 1 there is a (3g − 3)-dimensional
family of such varieties.
One tool used frequently in study is the canonical line bundle, denoted
KC , which looks like f (z)dz locally. Another commonly used symbol is H 0 ,
which is the set of polynomials of (deg(f ) − 2) on C. For instance H 0 (KP1 ) = 0.
In general H 0 (KC ) = Cg , where again g is the genus number. For g ≥ 2,
3KC is “very ample”; that is, the sections of H 0 (3KC ) give embeddings of
C ,→ PN and can be written locally as f (z)dz 3 . The Riemann-Roch Formula
4
gives H 0 (3KC ) ∼
= C5g−5 = {f0 , ...f5g−6 }, and the map we get C ,→ PN ⊃ CN
sends
f1 (x)
fN (x)
x 7→
,··· ,
.
f0 (x)
f0 (x)
I want to generalize all of this to higher dimensions, which is much harder.
For dim X = 2, we start with P ∈ P2 which corresponds to E ∼
= P1 ⊂ X
and is called the “blow-up of a point”. Blow-ups are essentially surgeries which
take lines through a point and spread them out. Birational equivalence in dim
X = 2 is generated by these blow-ups. In other words, X1 and X2 are birationally equivalent if there exists a series of blow-ups on each of them that will
end with the same set. This contrasts dim X = 1 in which two varieties are
birationally equivalent if and only they are isomorphic. For dimension 2, the
canonical line bundle KX looks like F (z1 , z2 )(dz1 ∧ dz2 ) and the sections are
∨
denoted ∧2 TX
. Four cases in this dimension can be explored.
Case -1: If H 0 (mKX ) = 0 for all m > 0, then X is birationally equivalent
to P1 × C, where C is some curve.
Case 0: If max dim H 0 (mKX ) = 1, then there are four possible scenarios.
In the simplest case, X looks like E1 × E2 where E1 and E2 are genus 1 elliptic
curves.
Case 1: If dim H 0 (mKX ) = O(m), then the general fiber F has genus 1 and
F → X → C, for some curve C.
Case 2: If dim H 0 (mKX ) = O(m2 ), then “almost anything goes”. It is
understood that the characteristic
χ(KX ) = dim H 0 (KX ) − dim H 1 (KX ) + dimH 2 (KX ) ≥ 1.
But even the case when χ(KX ) = 1 is not completely understood.
This brings up another question: “Can one understand the case when dim
X = 3 and max dim H 0 (mKX ) = 1 (the analog of g = 1, dim X=1)?” It is
not known whether there are finitely or infinitely many such families. This is
very important in theoretical physics, namely it could say how many possible
universes there are in the string theoretical framework.
There was a striking success by Bombieri about 35 years ago in the case of
dim X = 2. The canonical line bundle 5KX (which looks like f (z1 , z2 )(dz1 ∧
dz2 )⊗5 locally) gives an embedding X 99K PN . A recent achievement of mine
elaborates on these ideas:
Theorem 8 (Tsuji, Hacon, McKernan, and Takayama) If dim(X) = n
and dim H 0 (mKX ) = O(mn ) there exists an integer m0 depending on n such
that H 0 (m0 KX ) defines an embedding of X 99K PN .
However, even with this amazing result, “embarassingly little” is actually known
about m0 . For instance in the n = 3 case, 43 ≤ m0 (3) ≤ 10100 , but m0 (4) ≤???
5
Tommaso de Fernex
For this lecture, we will let k denote a field such as k = C.
We begin with Lüroth Problem (1861), which states: if k ( L ⊆ k(x1 , x2 , · · · , xm ),
then is L ∼
= k(θ1 , · · · , θn )?
Some positive answers to the Lüroth Problem have been obtained. Here
is one of them:
Theorem 9 If k ( L ⊆ k(x), then L ∼
= k(θ).
Sketch of Proof:
Let λ ∈ L\k. We then have k ⊂ k(λ) ⊂ L ⊂ k(x) with k(x) an algebraic
extension of L since the transcendence degrees of k(λ) over k and k(x) over k
are both 1.
Let p(t) = tr + a1 tr−1 + · · · + ar (with ai ∈ L ⊂ k(x)) and f (x, t) =
g(x)
b0 (x)tr + b1 (x)tr−1 + · · · + br (x) ∈ k[x, t] (with ai = bb0i ). Also, let θ = ai = h(x)
where g and h are coprime. Note that ai is not a constant.
g(x)
Then we have g(t) − θh(t) = g(t) − h(x)
h(t). This equation has a root
x = t, which implies p(t)|(g(t) − θh(t)). Then we have h(x)g(t) − g(x)h(t) =
p(x, t)f (x, t). Here we assume degx f = s which implies that degx p · f ≥ s. If p
is constant, then degx f = degx p · f . Now we turn to geometry. Perhaps we want to find all rational solutions to
x2 + t2 (x − 1)2 = 1. This equation is equivalent to (t2 + 1)x2 − 2t2 x + t2 − 1 = 0.
Thus, we obtain
(
q = (1, 0),
x=
2
−2t
pt = ( tt2 −1
+1 , t2 +1 )
which is a rational parameterization.
Let us work inside C2 ( or C). The complete picture is: C = {x2 +y 2 = z 2 } ∈
CP = C3 \{0}/ ∼. Here the quotient space is defined by the following equivalence relation: v ∼ v 0 if v = λv 0 for some λ ∈ C∗ . This is also a compactified
parameter space. Here is another example:
2
C ∪ {∞} = CP1 −→ C
t 7−→ pt
∞ 7−→ q
More generally, we can take X ⊂ CPr , a complex manifold defined by homogeneous polynomials. A typical example is X = {f (x0 , · · · , xr ) = 0} ⊂
6
CPr , wheref is a homogeneous polynomial. (In this talk, we only consider f (x0 · · · , xr ) =
xdr + · · · + xdr ).
We say X is rational if there exists a map:
CPn
X
∪
(n = dimC X)
∪
∼
=
U−
→ V
(birational map)
Similarly, X is unirational if there exists a map:
CPn
X
∪
∪
U V
A case of this is X = {x20 + x21 + x22 + x23 = 0} ⊆ CP3 , in which case
CP2 = {lines in CP3 passing through q}
X
“general line” l → p ∈ l ∩ X
∼
=
CP2 \{tangent lines} −
→ X\(l1 ∪ l2 ).
We can consider a complex manifold X ⊂ CPr
K(X) = {rational functions f :
X → C}. If K(X) ⊂ C, then K(X) are constant functions.
The idea is: {complex manifolds X ⊆ CPr , r ≥ 1 } / {birational equivalence}
→ {finitely generated field extensionsK ⊇ C} / {isomorphisms}.
Two examples are K(CP1 ) ∼
= C(x) and K(CPr ) ∼
= C(x1 , · · · , xr ). Here X is
∼
rational if and only if K(X) = C(x1 , · · · , xn ), where n = dimX. Xis unirational
if and only if C ⊆ K(X) ⊆ C(x1 , · · · , xm ), where m ≥ dimX.
The Lüroth Problem can be restated as: “Do we know whether X is rational
provided X is unrational?” We divide this problem into different cases.
We first revisit dimX = 1. Suppose X is a complex complete curve. Then it
is a Riemann surface and also a topologically orientable compact surface. Again,
let g be the genus of X. Then if g = 0, X is rational. On the other hand, if
g 6= 0, then X is not rational
Now suppose X is unirational and g ≥ 1. Suppose there is a surjective map
ϕ : CP1 → X. Let g0 be the genus of CP1 , then g0 = 0. By Riemann-Hurwitz
formula, we have 2 = 2−2g0 = degϕ·(2−2g)−ramification number. Therefore,
deg ϕ = 0.
We now turn to dimX = 2; let X be a complex surface (a 4-dimensional real
manifold). In 1896, Castelnuovo gave the criterion for rationality of complex
7
surfaces in terms of certain “genera”. (If dimX = 1 then X is rational if and
only if g(X) = 0.) One deduces that if X is a complex surface contained in
CPr , then X is rational if and only if X is unirational. This is the point to the
Lüroth problem.
For dimX ≥ 3, the first counterexample to the Lüroth problem for k = C
was constructed in the 1970’s. In dimX = 3 one counterexample is Xd =
{xd0 + · · · + xd4 } ⊆ CP4 , although this is hard to prove.
Here are the known results for the Lüroth problem for hypersurfaces in CP4 :
d
1
2
3
4
d>5
Xd
unirational
√
3
CP
√
quadratic
√
cubic
√
quartic
×
8
rational
√
√
×
×
×